A CATEGORY OF TRANSFORMATION 
GROUPS IN THREE AND FOUR 
DIMENSIONS. 


INAUGURAL DISSERTATION 


Submitted to the Philosophical Faculty of the University of Leipzig 
for the Degree of Doctor of Philosophy, 


BY 


JOHN VAN ETTEN WESTFALL. 


ITHACA, N. Y. 
ANDRUS & CHURCH 


1899 


VRE 


‘Written under the guidance of Professor Sophu 
at the University of Leipzig. z 
Day of examination: July 26, 1898. 


4 o) 


INTRODUCTION. 


Riemann, in his well known publication on the 
‘“‘ Hypothesen, welche der Geometrie zu Grunde liegen,” 
started from his definition of an arc-element in a 
manifoldness of n dimensions, 


ds ==> ae i 
in which the quantity under the radical is always posi- 
tive and then proceeded to the development of the con- 
ception of constant measure of curvature, which is per- 
haps the substantial result of the publication. Helm- 
holtz on the contrary goes still further back and at- 
tempts to found Riemann’s assumption. He makes his 
own assumptions and, from these as a base, he attempts 
to prove that all the transformations possible under his 
conditions have the invariant 
Sod cx Clore, 

“ze in an article in the “ Leipziger Berichte,” for Oc- 
tober, 1886, notes inaccuracies in the development and 
points out that, under one interpretation of Helm- 
holtz’s axiom of unrestricted motion (freie Beweglichheit), 
his monodrom axiom is entirely superfluous, while un- 
der another interpretation, even all the axioms would be 
insufficient to determine the groups, which preserve the 
geometric qualities of a rigid body in space. Lie, in a 
later article in the same publication,’ proves rigidly the 


1 Leipziger Berichte, Oct., 1890. 


isk yaaa 


truth of his statement made in his article of 1886. He 
proceeds from Helmholtz’s assumptions, with the excep- 
tion of one, namely, the monodrom axiom and deter- 
mines all the groups, satisfying the given conditions. 
Under the most general interpretation of the axiom of 
‘unrestricted motion,” he gets, besides the groups of 
Euclidian and non-Euclidian motion, five others. Kow- 
alewski, a pupil of Lie, has in his inaugural disser- 
tation’ extended Lie’s investigations in space of three 
dimensions to that of four and five. Besides the Euclid- 
ian and non-Euclidian groups, he finds three others that 
satisfy the condition of unrestricted motion. The dis- 
cussion of these eight groups, or more particularly the 
one-parametric sub-groups, is the chief aim of this dis- 
sertation. ; 

The behavior of points under certain special con- 
ditions is in some cases most remarkable. Some- 
times all the points of a surface remain at rest and some- 
times not, depending upon the choice of the points we 
hold stationary. In some cases too, we find all the 
points of a surface invariant wherever we may choose 
our points. Then, too, in space of four dimensions, we 
find one group has in some cases closed path-curves and 
in others not. 

To give Lie’s development in its entirety is of course 
out of the question, but at the same time, in order that 
we may have a proper insight into the groups, a short 
sketch is necessary. 


1 ‘Uber eine Kategorie von Transformationsgruppen einer vierdim- 
ensionalen Mannigfaltigkeit.’’ 1898. 


CHAPTERIL 


SKETCH OF LIE’S DEVELOPMENT OF THE GROUPS 
SATISFVING THE AXIOMS OF HELMHOLTZ, 


Lie sums up the Helmholtz axioms in the following 


words :' 
Bet 
Yi 6 OX) WV) oy aie ea 
Woe (EV Ay ene tec 
Zp VY NEO ee tne eee 


be a set of real transformations in space of three dimen- 
sions under the following conditions : 

(A) The functions -, ¢, w are analytical functions 
of the variables and the parameters. 

(B) Two points shall have one and only one invari- 
ant in the group. 

(C) There shall be unrestricted motion in space, that 
is: the point x, y, z, can be transformed into every other 
point in space. If we keep x, y, z, fixed, then a second 
point can take o” positions. Hold two points station- 
ary, then a third point can take o!' positions. Finally, 
if we hold three points stationary, then all the points 
in space remain stationary. 

(D) If we hold two points stationary and transform 
the remaining points in all possible ways, then the 
points, after traversing a finite distance, shall return at 
the same time to their initial positions, 


1 Leipziger Berichte, Oct., 1890, 


slg sn 


That this set of transformations under the conditions 
A, B, C form a group with six parameters is evident. 
We have then the problem, to find all the six parametric 
groups in space, which are defined by real analytical 
equations and for all real points satisfy the conditions 
Brands: 

The six equations: 


5 8 
ers al fh (a Yu Z,) ea Winn e SiG ete b. C5 Yo» 25) mas as 
Xx, 02, 
(k = Ge 2. e . Gk 


can have one and only one solution. ‘Therefore 
= DS Se OR va or) eet a 


while the 5 rowed sub-determinants do not all vanish. 
We can put this criterion in another form. We mul- 


tiply the W,- by such quantities ¥, ... <2 apaegmee 
the co-efficients of ig sf, Aa in the expression y¥, W, -+ 


.+yW, W,f, shall be equal to zero. We obtain 
three equations in the y’s with co-efficients depending on 
X1, Yi Z,, which we can solve for-three of the Ws.Guuie 
other three are indeterminate. We choose three of the 
many expressions 

Vif = 2, WF 


so that no relation 


0, (%,Yp%) Vip +o, Vif +o, V,F =O 


exists, and set 


8x, if dy ioe 


2 


ane as 4. 7») SF aa oF 


ly 
The determinant A, —2=+ {7',@, vanishes with A, 
and the two rowed sub-determinants vanish with the 
five rowed sub-determinants of A.. The geometric sig- 
nificance of this is, that a pair of points can not be trans- 
formed into every other pair but only into o’ pairs, and 
that is exactly the significance of the condition that two 
points have one and only one invariant. We have then 
the following criterion: If three independent infinitesi- 
mal transformations, Y,¢=ap+Bq+y¥r, of a six- 
parametric group X,¢ .. . X,F, leave a point invari- 
ant, then two points have one and only one invariant, 
when the determinant of the sub-group vanishes identi- 
cally, while the two rowed sub-determinants do not all 
vanish. 

It is evident, that after holding one point x,, y,, z, sta- 
tionary, all the other points move on o’' invariant sur- 
faces. In the case of the Euclidian motion, these sur- 
faces are the oo’ spheres with the centre x,, y,, z, which 
we will call pseudospheres. In the general case, we 
will therefore call the o' invariant surfaces pseudo- 
spheres, with the centre x, y, z, All these satisfy the 
three partial differential equations, 


Yer, Gk) eae) 


only two of which are independent. It is evident that 
a, B, y are not independent of x,, y,, z,, else there would be 
only oo’ pseudospheres, which would violate the axiom 
of unrestricted motion. We see then that there are 
at least «” pseudospheres in space. It is also evident, 
that there is not even one partial differential equation 


NS ge 


Af =0, whose coefficients are independent of x, y, Z,, 
which is satisfied by all the pseudospheres in space. It 
follows easily then that among the infinitesimal trans- 
formations, no two occur, which satisfy the relation 


$, Sif + $, X, f= 0. 


If we hold a point stationary under our six-parametric 
group, then the line-elements, through the point, will be 
permuted among themselves by means of a projective 
group with at the most three parameters. We know 
also that a projective group in a plane, with not more 
than three parameters, has either an invariant pcint or 
an invariant conic. If we regard the co-ordinates of the 
line-elements dx, dy, dz as homogeneous co-ordinates 
in a plane, we come to the conclusion that our six-para- 
metric group leaves either a line-element or a cone in- 
variant. In the former case our group is imprimitive 
and leaves a set of co” curves invariant. In the latter 
case our group is primitive and has the invariant differ- 
ential equation: 


a,, dx’ + a,, dy’ + a,,dz’+ 2a,,dx dy 
pee GyGZ rig o, eda 


All the six-parametric primitive groups satisfying our 
conditions have been determined in an earlier publica- 
tion by Lie and are similar either to the Euclidian or 
nou-Euclidian groups by means of a real transformation. 
If the six-parametric group is imprimitive then it can 
occur that o° surfaces remain invariant and it is in 
that case easy to prove that the surfaces are interchanged 
among themselves by means of a three-parametric group. 


For if we introduce the invariant ¢(x,y,z) as a new 
variable x in our group, then the coefficients of all the 
p’s will be simply functions of x and the shortened in- 
finitesimal transformations €, (x) p form a group in one 
variable, which transforms the surfaces, at the most, bya 
three-parametric group. It is easy to see that the short- 
ened group is neither one or two-parametric, for in the 
first case there would exist only o' pseudospheres in 
space and in the second case two of the infinitesimal 
transformations would have the same path curves, both 
of which are excluded. We have the result then, that 
the set of surfaces are transformed by a group with three 
parameters. 

We now go back to the case where our group leaves 
a line-element through a general point invariant. There 
exists then an invariant simultaneous system, 


dx dy az dz 


Bixty zy mn Cxuy, Zz) Ry oneyaae 


That is: the set of curves determined by the solutions, 


$ (x,y,z) =¢, and ¥ (x,y,z) =¢, 


are invariant under our group. If we introduce these in- 
variants, as new x and y in our group, we obtain, 


Reh eee Pt HX, ¥) de, Ca) t. 
The shortened transformations, 
Vif = & (X,Y) p + 0, (x,y) q 


form then a group which shows how the o? surfaces 
are transformed. Ifthe group had less than five par- 


ameters, there would exist two infinitesimal transforma-_ 
tions with the same path curves, which is excluded. 
The set of straight lines is then transformed either five 
or six-parametrically. Is the group Y,¢ primitive and 
s1x-parametric, then it has the form 


Pp; q, xq, >. OF y q, y p. 1 


If it is imprimitive, then it has at least an invariant set 
of curves ¢(x, y)=c. ‘This, considered as an equation 
in space, represents a set of surfaces which is invariant 
for the group X,¢ and is transformed three-parametric- 
ally. ‘The curves ¢ (x, y) =c are therefore transformed 
in the same way and we have only to find the groups Y, F, 
which transform the invariant set of curves ¢ (x, y) three- | 
parametrically. The six parametric groups which con- 
form to this condition are the following : 


Dy: Dre ae i aC Le 
q, Reqhorx Ge pox p -+y ay Sposa ae Lil. 
yee Geo, a iy Gh. xX Diao ee Tye 


If the group Y, ¢ is primitive and five-parametric, it has 
then the following form: 


P, 4, xq, Xp—yq, YP. V. 
If imprimitive : 
PG XG, 2xptyq, x ptxy4q. VI. 
The groups sought for have the form: 
ee Vine eee 


where the Y,¢ have one of the six forms above. By 


bracket expressions we determine the @’s in all the cases 
and then examine the groups to see that they fulfil our 
conditions. The groups, obtained from the shortened 
groups I and III, do not conform to our conditions and 
from the others we obtain the four following: 


emcee sv Cart kor, ie 
et 2aet wl ¥ dec. 2k v1, 


Pp 4, xq+r, x'q+2xr LT 
xptyqtAr, x*pt2xyqt2(Ax+a2y)r. 


Heavier x T, 2 oT; Bile 
Peete YG YP: 


Pp, q, Tf, Ue aan Re rH. Og mane WF Live 
xp+xyqtayy't. 


As yet the problem has been solved only under the 
condition, that the quantities x y z are complex. We 
wish however to find all the real finite groups which 
satisfy our conditions. Ifa real group satisfies the con- 
ditions in regard to the invariants of two or more points, 
the group obtained by considering the variables x yz as 
complex quantities will also satisfy the conditions and 
is therefore six-parametric, andis by a real or complex 
transformation similar to one of the groups already 
formed. It is necessary therefore to find all real groups 
which are similar to the above mentioned groups by 
imaginary transformation. The groups found similar 
to II, HI and IV are similar by means of a real substi- 
tution. Further, all groups similar to I by means of 


— 14 — 


real or imaginary transformation are similar by means. | 
of a real substitution, except the following: 


Pad ox Deed 

yp X de NF 
(x’—y’)pt+2xyqt(2k,x—2my)r 

2x y Pp Peer ie ak Viet 2 ee 


which is similar by means of the substitution 


X,=xtiy y,=x-ly 4,= fe 
; : *  k +im 
in which, both k, and m can not vanish, and in which 


[ioe TE 0) 
kc AEE, 
el 


We have then this group V to add to our list. 

Kowalewski by a similar discussion found besides. 
the Euclidian and non-Euclidian groups in space of four 
dimensions the following three, which satisfy the condi- 
Hons. A. Bate 


Py — ¥%_P3, Pe ar X, Ps, %,P2 %, Py — ¥, Po. %_Pi; 
X, De tee eRe py oy UO Daa KE ee Vie 


Pi» Po Ps, %, Po — ¥3 Pi» %2P3 — Xz Pos 
X3P; — X, Ps, WER: 
Eh. 2S D7 BU, 2x,U Se 


Pi, Pr Ps, Xi P2— ¥: Py, X2P3 + X3 Py, Xs Py + X, Psy; 
U, 2x, bio (xy + so X, ) Pir 2 x,U =e Xai ea Poy 
2,00 CS xe ee pe VIII. 
where 
A Dy che Daisies se Pata a 


xy 7K, KS 


GA Pires ble 


DISCUSSION OF THE GROUPS IN SPACE OF THREE 
DIMENSIONS. f 


Definition.—A general point under a transitive r- 
parametric group in n-dimensions is one for which all 
the n-rowed determinants of the matrix do not identi- 
cally vanish. A general point, under an instransitive 
group, is one for which the largest non-vanishing deter- 
minants of the matrix do not all vanish. 

Definition.—A group is systatic, when by holding a 
general point fixed, all the points of a certain manifold- 
ness remain at rest. All of the groups under discussion 
are systatic, because all the transformations are per- 
mutable with r. We can see this directly, for by hold- 
Poem come point. x = y =z —0,..which is” a pen- 
etal point, all the: points xo y =o zZ==Z remain at 
rest. ‘There are therefore o' points which can not take 
Smeposttiouseaiter Yolding xy —az =o xed. On 
the other hand a general point can do this. 


GROUP I. 


Dy, oGsee Der, Vet ekis, 
x'p-2xr, yqite2kyr. 


The invariant of two points has the form: 


[=2,+ Z,— log mao Xo k log ‘Yar Ne) 


io 

The three-parametric sub-group obtained by holding 
the point x sy == 7 =='0 fixed iis : | 
yq—kxp, x*pt+2xr, yqt2kyr. i 


The group is of course instransitive and any point, for 
which both x and y are not equal to zero, is a general 
point. If we hold the point x,, y,, z,, (X)3= Ojameme 
fixed the resulting sub-group has the form, 


(3 Xo Vo X k Yo Bs) p+ (y’—y Yo) q+ (7 z Ao ane RY. =r. 
c pe 


2 


The differential equations for determining the path 
curves are these: 


dheie: x’ (x ,— x’) : 
_= | o 

dt a x, 

CS meaay enyates 

ape ae: (y Yoo 


where [x’],_,—x, etc. We get then from tigse aaa 


equations and that of the pseudosphere 


x/= x ek Yot | Nicos aes 
eYot 


12 / 
ZZ + log * +k log =. 
x bs 


However large t may become it is impossible for a point 
to return again to its initial position. By setting the 
coefficients of p, q and r equal to zero, we see that the 
only points remaining at rest are: (0, 0, z) and (X,, Vy Z). 
If however we choose x, = 0, or y, = 0, which we obvi- 
ously have a right to do, as in both cases the point is 
general, we obtain much simpler forms for the path- 


curves and, instead of all the points of two curves re- 
maining at rest, all the points of a certain surface re- 


main stationary. 
For x, = 0, y # 0, the sub-group has the form 


xp +2 xT. 


Here it is obvious that all the points of the surface 


X =oOremain at rest. The equations of the path-curves 


are easily found to be 


Stim? Sf 
, 
'=z+2log~. 
ss 
Each of the surfaces of the set y = const. is invariant. 
For y, = 0, X, # 0, the sub-group is 
yqt2kyr. 


All the points of the surface y = o remain at rest and 


the equations of the path-curves are 


ns y 
z=—Z2+klog—. 
Me 


Fach surface of the set x = const. remains invariant. 


pone) 18 ke 
GPOUP II. 


Dee Cre csr io eae 
Spey Cee po 2 XK yed ach A 
The invariant of two points is: 
l=z+2,- = Aide: (x,— x)’ ae EE 
xy ae 
By holding the general point x = y = z = 0 fea 
get the sub-group 


xpt+(y—Ax)q, x’q+e2xr, 
x p+2xyq tayo x) r. 


Examination of the matrix shows that all points, with 
the exception of those for which x and y are both equal 
to zero, are general points. If we hold the general point 
(0, Yor Zp) Yo # O, Z # O fixed, we obtain the sub-group 


x’q+2xr. 


The finite equations are then: 
x= xy ay Se bee 7 ee 


Here all points of the surface x = o remain fixed 
and each surface in the set of surfaces x = const. is in- 
variant. If on the other hand we take our point outside 
of the surface x = 0, there exists no surtacesavnen 
whose points remain at rest. 

The sub-group, obtained by holding x,, y,, z, (x, # 9, 
y, * ©) fixed, is: 


x (x — x,) p+} (Ax—y) pone’ Vt 


Xo 


+1 2(y+Ax)— 220.1 Saale 


x 


te) 


and wé see that only the points (0, 0, z) and (x, Yy Z) 
remain at rest. 
Grovp III. 

Rises. 4 fe r, f, 

So re gee De a ee 
The invariant of two points is: 

LS heme 7 Hn wd SA themed 054: 
The invariant sub-group obtained by holding the gen- 
Beta OMCs nye 7 OL xediis:: 
<0 yan a Ve ava HT 


By examining the matrix we see that all points, except 
those for which x and y both are equal to zero, are gen- 
eral points. 

The one-parametric sub-group, obtained by holding 
the second point X,, Yo) Z) (X) # O, Y) # O) fixed, is: 


CAB) jnt (2 Ja 
Yo Yo Yo 
from which the finite equations are: 


x’ =x + *o ese t, 
Y oe Yo 
yi =yt(x—%y jt, 
Yo 


Z=Z'. 


Here, in contrast to the other groups that we have 
examined, all the points of the surface x y,— x,y =o 
remain at rest. 


meio) a 
If we take x, = 0 the sub-group takes the simple form 
aie 
with the finite equations : 
SO eer oy meet Kn ons 


and the path-curves are parallel to the y-axis. All 
points in the surface x = o remain at rest. If we take 
Y, = 0 we get the sub-group x p with path-curves par- 
allel to the x-axis. In this case we see that, wherever 
we may choose our general point, all the points of a cer- 
tain surface are invariant, while with the other two 
groups that was not the case. 


Group IV. 
Dyoid (oe Xs ie tye KD vi, 
Ro ey Oat ay OTs 
The invariant of two points is: 


(Cy, We Vo)e 


Rt Beers Bras “ 
20x) 


The sub-group determined by holding the general point. 
x= y = 72 — 0 fixed.is thefollowine: 


SoC 2x Dis 3G, IY; 
x*p+xyqttiy’r. 
If we examine the two-rowed sub-determinants of the 
matrix we see they do not all vanish except for the 
values x = 0, y = 0, so that we can choose any point 
for which both x and y are not at the same time equal 


ome ORY See 


to zero. We shall see that in this case wherever we 
may choose our second point there will be no surface 
the points of which all remain at rest. The one-para- 
metric group formed by holding x,, y,, Z)) (X)# 0, y)# ©) 
fixed has the form : 


2x x, 2 x (2 eee ee BL) 
( = )p+ )at( pds hi 


Xp 


Morex, —O the sub-group is: 

P(X Vi Vg) eh ee Wyo) it 
and for y, = 0 the sub-group has the form: 

ae) x) PE (XY ex yd eye. 
In each of the three cases the only invariant points are 
the ones on the two lines which are parallel to the z-axis 
and pass through the stationary points. If we determine 
the finite equations of the group, we find that asin all the 
other cases the path-curves are not closed. In this case 


the expressions are very complicated, but we need only 
to determine x to see the truth of the statement. 


GRovpP V. 


In this group, as we have seen, both m and k, can not 
be set equal to zero. If one of them however be set 
equal to zero, the other can be made equal to one and 
for convenience of discussion we can write our group 
‘in the form : | 
PN Fis al ney et 
U, 2xU-—Sp, 2yU—Sq, 
U=xpt+yaq+ir, S== fy. 


The invariant of two points is: 


| y= x) Gi— v2) fev ets) 


and the sub-group, obtained by holding the general 
Point = y ==7 no xedus: 


XQ 12x — Sp, .2y US sig 


For the second point we may choose any point for 
which both x and y are not equal to zero. However 
ingeniously we may choose our points it is impossible 
to bring the one-parametric sub-group to a very simple 
form. In four dimensions and in fact in n dimensions, 
where we find groups analagous to this we shall see that 
for any value of n > 3, we can always choose our points 
so that the one-parametric group shall take the form of 
a rotation. The one-parametric group is in the case 
aU eos FC 


pane: = sane 
| uf oe + vA Xp ty, p 


297 XY x 2 2 2x,Y¥— 2y,x 
+}x- 0 — 8 (x? — ; +( 0 0 yr 
ie San fein. tg vi yaya x ae 


from which the finite equations are the following: 


(x, tiy,) (& +i1y) 


x +1iy = 
fx,— x +i(y,—y)}e* +x +iy 
x? + y” 
A May abies Oe (gh gt aural Sa 
A & x’? + y’ 


However we may choose x, and y,, the only real points 


1 Theory of Transformation Groups. Vol. III, p. 459. 


remaining at rest are those of the straight lines through 
the points (0,0,0,) and x,, y,,Z). parallel to the z-axis. 
From the finite equations it is easy to see that the path- 
curves are closed. 

Of all the one-parametric sub-groups there is only 
one, namely, that of group III, which leaves all the 
points of a certain surface invariant wherever we may 
choose the same general point x,, y,, z,. On the other 
hand there are two, namely, those of groups I and II, 
which have this characteristic, in case we suitably 
choose our general point x, y,)Z, Finally, there are 
two which under no conditions leave all the points of 
a surface invariant. 


CHAPTER, IL 
THE GROUPS IN SPACE OF FOUR DIMENSIONS. 


The only groups in space of four and five dimensions 
of those determined by Kowalewski, which satisfy 
the axiom of unrestricted motion in either sense be- 
sides the Euclidian and non-Euclidian groups, are the 1m- 
primitive groups we have numbered VI, VII and VIII, 
and two groups analogous to VI and VII in space of five 
dimensions. Kowalewski has shown that groups 
analogous to VI and VII exist in space of n dimensions. 
Further, from analogy between groups VII and V, he 
draws the conclusion that the one parametric sub-group, 
obtained from VII by holding three general points sta- 
tionary, has closed path-curves. By suitably choosing 
our general points we shall prove the truth of this state- 
ment, without being compelled to.resort to the long inte- 
gration which would be necessary if we chose our points 
entirely general. | | 


Group VII. 


Pir Por Ps, %1 Pp ~ X, Py, *X_ Ps — Xs Py, 
X3 Pi — X, Ps: i 2x Wp 
2x,U —Sp,, 2x, U —Sp,, 


U=x, PtxPtaPpt+tpyHSs =. + eee 


The invariant of two points is: 


J gh Sa y,) 
I= | (xX, se! Na te Cx: = Vo or (xy he 


Our six-parametric group obtained by holding the 
general point P, = (0, 0, 0, 0) fixed is the following : 


X,P. 7 X, Py, X, Ps — X3P., X3 Py — X, Ps» 


2x,U—Sp,, 2x,U—Sp,, 2x,U—Sp,. 


The matrix is in this case: 


— x, <7 O oO 

— x, Se O 

2a O — x, fe) 
x7? — x? — x,’ 2X,xX 2x,x 2x 
1 2 3 1 4 1 3 1 
2 x%*—x7—x? 2xx 2x 
xX, X 2 8 1 2 3 2 
2 2X,X x, —x7?—-x,” 2x 
Xs x 3 “2 3 1 2 3 


For no values of the variables except x, = 0, x, = 0, 
x, = 0, do all the three-rowed determinants vanish. If 
we choose for our second point P, = (0, 0, z,, Z,), where 
Z, # O, we obtain the sub-group: 


X, P, — X%. Py, 2 Ut Dae Oe Diam, Dy) s 
Zp GAB tel Se ig 0) ody Ap b> Sees hu 9 


Hor, our third ‘point we may take Py=— (0; 0; a’ ;, 0), 
where z, # z’, # 0, for in this case the matrix is: 


O O 10) 1@) 
— z”,+ 2,2’, O O fe) 
O — Zz", +.2,2', O O 


me) 6 


in which all the two-rowed determinants do not vanish. 
We get then our sub-group in the form: 


X, P, — X_ Py. 

In space of five dimensions the result is exactly the 
same and, if we examine the analogous group in n di- 
mensions, we see that it is always possible to choose our 
general points so that our one parametric group shall 
take the form of the rotation: 


ae he Diy ee oat 


From this fact alone, however, we can not draw the con- 
clusion that our sub-group always has closed path-curves, 
wherever we may choose our three general points, as we 
shall see in the case of group VIII. 


Group VIII. 


Py, Pos Pg, %1 Po —~ Xo Pir Xe Ps 55 X3 Po» 
xy Pi teks u; 22S pr 
2 = pi. 2 Kr Oy. 


2 


= X) Peete Sy Pst ky Pe Pa S= x + x, ae 


The invariant of two points is: 
: : : site (x, ne Yu) 
I= |v + Gy Gy) be 
The six parametric sub-group obtained by holding the 
general, poimt Pi=— (0, 0,°0, 0) fixed is: | 


X,P.— ¥, Pi, * Ps a X3 Py, Xs Py AP X, Ps, 
2x,U—Sp, 2x,U—Sp,. 2%,U toe 


The matrix is in this case: 


— x, X; O fe) 

xe ae O 

x, Oo eh O 
x, —x,4+ x,’ rd pg a BEKO Xs 7190 
2x, x; xX, — x, +x,’ PENS gb 2X, 
oie PES ED.D X, +x, +x,’ SS. 


Ponno wales of, the variables except x, == x, = x, ==.0, 
do all the three rowed determinants vanish. If we 
choose for our second point P, = (z, 0, 0, z,), where z,#o 
we obtain the sub-group: 


me Pact Xs Ps Dee it) aes CXED eae, P,) Z, 
Cee Ueat et) ai CS, Dy cts bey Zy. 


If we choose for our third point P, = (z’,; 0, 0, 0), 
where z, # z,' # 0, we have the matrix: 


fe) fe) fe) 
12 1 
fe) ara AS pig eeY A fe) 
= 12 / 
fe) O Come. Dy 


which shows that our point is general. ‘ We obtain then 
for our one parametric sub-group: 
X, Ps + Xs Pa 


which plainly has not closed path-curves. If, on the 
other hand, we choose for our second point 
P, = (0, 0, Z,, Z,), where z, # 0, we get the sub-group: 


X,P, — X, Pi Ser eal Dy rie carseat) , 
2x,U+Sp, + Z, (X2 ps + X; p,). 


pes J) 


Further, by holding the point P, = (0, 0, z’,, 0) where 
Z, # Z', + O, we obtain for the sub-group, the rotation : 


X, Py — X,_ Pi- 


We see then that in this case the path-curves are closed, - 
while in the other they are not. This has its explana- 
tion in the fact that in general a triplet of points can 
not be transformed into every other triplet by means of 
a transformation of the group. Each triplet has three 
independent invariants; and it is possible to transform 
one triplet into another only when the three invariants 
of one are equal to three invariants of the other. If 
then in one case the path-curves are closed, they will be 
so in the other and vice versa. | | 

Let us examine now our group VII and see if it is pos- 
sible to choose such a point-triplet that our path-curves 
would not be closed. The special point-triplet we used 
in determining our sub-group was: 


P,=(0 fe) O el) 
P,=(0 O Zi jt) 
P, = (0 fe) ae O ) 


The invariants obtained by setting the co-ordinates of 
these three points in the general form for the invariant 
of two points are: 


For all positive values z, z’, and z,, the invariants are 


positive and never equal to zero. They are further in- 
dependent with respect to z,, z’, and z,, as we see from the 
functional determinant. If we choose any real positive 
values whatever and set them equal to the expressions 
above, we can solve for z,z’,z, If we set 


itis 2 2 a 
2 —— , 2 n2 2 
Z,e=a°, z’,=b’, (z,—2,7;e=ec 


and solve, we obtain the real values: 


ee baie a atts b? 
jae heey A Rey Fath hares f 
I—c I—c 


From the form of the general invariant, we see that all 
triplets of general points have positive invariants which 
can never be equal to zero. If, therefore, we choose any 
triplet of points whatever, we can always’ determine 
Zs Z's, Z, 50 that the invariants of the one triplet will be 
equal to those of the other and we can therefore trans- 
form one into the other by means of a transformation of 
our group. In our special case the path-curves were 
closed and we therefore know that they are closed in 
general. 

The one-parametric sub-group of group VIII has 
closed path-curves, as we have seen, when we hold fixed 
the triplet of general points: 


P,=(0 fe) O O:) 
PR=(0 0 % 4) 
P, = (0 O A Ta 


‘The invariants obtained from setting these values in 
the general form of the invariant are: 


Lin 


Bie aah esa: 

ase au gare ye 2) 
I= Z 3 

WAT pt We ape 2 
I,, = (Zs AS € 


The expressions are all negative, and never equal to 
~ Further, 
are independent with respect to these quantities as we 


zero for the possible values of z,, z’,, z 


see from their functional determinant. By the same 
reasoning as before, we see that if we choose any triplet 
of general points whatever whose invariants are all neg- 
ative, we can always determine z,, z’,, and z,, so that the 
invariants of the one triplet will be equal to the invariants 
of the other and we can therefore transform one into 
the other by means of a transformation of the group. 
We have then the result that all groups, obtained by 
holding stationary a triplet of general points whose inva- 
riants are all negative, have closed path-curves. The 
question then arises, whether these are all the sub- 
groups that have this characteristic. The question can 
be answered in the affirmative, for we know that group 
VII is similar to group VIII by means of the transfor- 
mation x’, 1 x,, and x, p, — x, p, is the only transfor 
mation of the group which remains invariant under this 
substitution. The invariants of the three fixed points 
are transformed into 


IL,=—-Ze 4 1,=—z,;, L.=—(@ —Z) eae 
We see then that only those point triplets whose invari- 
ants are all negative give us groups with closed path- 
curves. 


We have now only the group VI to discuss. This is 


analogous to group III in space of three dimensions and 
has not closed path-curves. ‘This I have not been able 
to prove as in the case of the two groups above by an 
ingenious choice of the general points. We can how- 
ever so choose our points that the sub-group appears in 
a very simple form. 


Group VI. 


P; — ¥,P3, Pe 2 X,P3, P3, XX, Po, 
X, Py — X, Py, X%, Pi, Xs Ps =e 
Xs, pi; — X, pe Xs; Pe a x) Wy; X, U, 
eae xy Py i Xa De 1 Xs Dette De 


The sub-group obtained by holding the point (0, 0, 0, 0) 
fixed is: 
XP. %,P, ~ %,P2, Pir X3 Py — X, 1 
ey Doren Uy a, U 
As second general point we can choose (0, 0, Z,, z,) where 
Zz, # O, and we obtain the sub-group 


X,P,., %, Pi — X_ Py X,y Pi- 


As third general point we can choose (z,, 0, z,, z,) where 
z, # 0, and we get the very simple group 


X, Pi 


in which the path-curves are plainly not closed and run 
parallel to the x,-axis. All the points in the manifold- 
ness X,= 0 remain at rest. Group VI has its analogous 
group in n dimensions and it is easy to see that by suit- 
ably choosing our points we can always obtain the one- 


parametric sub-group in this simple form. It is impos- 
sible in this case to show by the form of the invariants 
that the path-curves are in general closed. To do this 
it is necessary to determine the sub-group that leaves 
the poiut-triplet :(0, 0, ©,'0), \(Z,, Z.) Zs) 2,), (29) 2 ee 
invariant and then calculate the finite equations of the 
group. In that case we would find that analogous to 
group III all the points of a plain manifoldness would | 
remain at rest. It is interesting to note that with all 
the groups, except V and VIII, the invariant of two gen- 
eral points can vanish without the points coinciding. 
If we interpret axiom C as meaning that after holding 
a general point fixed it is possible to define a space 
about that point, such that every point contained there- 
in can take o’ positions in space of three dimensions 
and o* positions in that of four,:then, as Lie simeuie 
publication of 1890 points out, these eight groups must 
be excluded and we have only the Euclidian and non- 
Euclidian groups. If however we give axiom C the in- 
terpretation that after holding a point stationary a gen- 
eral point can take o’ positions or o° positions as the 
case may be, then we must include the groups above. 
In this case we see that if we call the monodrom axiom 
to our aid, the groups V and VII would still have to be 
included. We.=see then that in one case the monodrom 
axiom is superfluous and in the other insufficient to de- 
fine the Euclidian and non-Euclidian groups. 


1 Leipziger Berichte, 1890. 


NicD AS 


I, John VanEtten Westfall, was born on the 24 of 
June, 1872, in Dresserville, New York. My schooling 
preparatory for the university I received in Ithaca, New 
York, at the close of which, I entered Cornell Universi- 
ty, from which I received the degree of Bachelor of 
Science in 1895. Directly afterwards I came to Ger- 
many where I have studied three semesters in Gottingen 
and three in Leipzig. I have heard lectures under 
Klein, Hilbert, Schoenflies and Sommerfeld in Gottin- 
gen; and Lie, Mayer and Volkelt in Leipzig. 


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